The application of thin-wall shell model in plate problem of syntactic foams

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1 Available online Journal of Cemical and Parmaceutical Researc, 14, 6(6: Researc Article ISSN : CODEN(USA : JCPRC5 Te application of tin-wall sell model in plate problem of syntactic foams You Liwen 1 and Zeng Ziun * 1 University of Science and Tecnology Liaoning, Ansan Department of Mecanics and Engineering Science, Pecking University, Beiing ABSTRACT Te composite materials wit vacuole inclusions are modeled by mesing matrix materials into entity units and inclusions into tin-sell units. Te element stiffness of te smoot closed tin-wall in plain-strain condition is formulated by Kircoff-Love teory. And corresponding to te model, te multi-point constraint conditions of te material interface are obtained. A representative volume of composite materials wit vacuoles inclusion is taken for example; te effective modulus variance during te interfacial debonding procedure is simulated. Te result sows tat te mesing sceme and te assembling sceme proposed are valid and te number of freedoms is reduced. Keywords: composite materials; vacuole inclusion; effective modulus; multi-point constraint; debonding CLC No.: O343.7 INTRODUCTION Composite material is a kind of common compound material. Troug embedding glass ollow inclusions in te matrix, it acieves [1],presented by Figure 1. Embedding ollow inclusions can make materials ligter and softer [][3], enance materials termal expansion effect [4] and absorbing capacity [5], and promote cusion performance [6]. Nowadays, it as been applied to aerospace, navigation, and logistics [6]. Figure 1,ollow microsperes syntactic foam Finite element modeling calculation plays an important role in discovering te mecanic attributes of syntactic foams [7]. For te researc of normal intension of fibrous materials, it can be simplified in accordance wit plane strain metod [8]. In terms of geometry,ollow inclusion belongs to tin-wall sell structure. For Finite element 1748

2 model of tin-wall sell, not only solid modeling, but also tin-wall teorem can be applied to [9]. In suc problems, bot inclusion and matrix are always segmented into entitative unites because of te special connection between matrix and inclusion tin-wall sell. Now, in order to describe te curve of tin-wall sell, set more unites along wit tickness wose dimension is small in tickness direction. And in order to avoid ill-conditioned stiffness matrix, te corresponding scale of unites in sell as better lessen, and simultaneously te scale of gridding is in smoot transition along wit te distance to interface wile degrees of freedom increase muc [7][8]. To reduce cost of calculations, te paper tries to de-dimension inclusions in according to classic Kircoff-Love tin-wall sell teorem, deducing te expression of strain and stiffness of inclusion under plate strain situation, discussing multi-points constraint relationsip between inclusion and matrix, and ten taking te equivalent modulus problem tat at te time syntactic foams debond partly from interface as an example for cecking.. Te stiffness inference of inclusion sell Because one side of inclusion tin-wall is air, it is almost considered as te prerequisite of only tangential normal stress. If we assume tat tin-wall curve is smoot and its slope is not tat steep, we can adopt te sell model of Kircoff-Love. Tis model equals to an arc tat meets plane s assumption in te case of strain plane, and only as positive strain along wit te direction of tangent (sown in Figure. Te normal commercial software does never fulfill tis plane strain unit [9]. In order to make programming convenient, tis article will make inferences of its corresponding expression under rectangular coordinate system,and ten acieve element stiffness matrix. Figure, microelement of te inclusion We can know easily from microelement sown in Figure tat te original distance of tangential section away from z point to neutral surface is: l = s (1 + κ z (1 κ is te original curvature, expressed as follows: κ = y'' x' x'' y' ( Te derivative is differentiation of arc lengt. Pay attention to te requirements of Kircoff-Love tat need to be small. If tin-wall microelements ave micro displacement ( u( s, v( s,te cange of surface witin tin wall is ( x( s + u( s, y( s + v( s,eac variable in formula (1 sould be: κ s = s 1+ u ' x ' + v ' y ' + u ' + v ' = s (1 + u ' x ' + v ' y ' (3 (( y '' v ''( x ' u ' ( x '' u '' ( y ' v ' κ = ( = x ' y '' y ' x '' + ( x ' v '' u '' y ' s s (4 1749

3 l s( κz 1 Te above derivative uses te prerequisite tat ( u / s, v / s = + (5 is a small value. Terefore, te positive strain of eac point wit te direction of tangentiality can be presented as: ε l l l = = ( s s + z ( sκ s κ s ( u ' x' y ' v' z ( x' v'' u '' y ' zκ ( u ' x' y ' v' = (6 Obviously, te first term in formula (6 stands for tensile strain mode similar to pole. Te second term stands for curving strain mode similar to girder. Te tird term describes tat fact tat te bend of tin wall will absolutely lead to te curving strain simultaneously wile stretcing. And we can realize tat wen te neutral surface of tin wall is straigt line, te strain mode expressed in formula (6 will degrade completely to be te combination of pole and girder. Te above formulae as second derivative of displacement ( u( s, v( s. Wile using finite element discretization, sape function sould be second derivative. In order to make bending moment conveyed, te first derivative of connection point sould be continuous. Terefore, te paper adopts te classic beam element of sape function called two nodes and four parameters to insert value, tat is: (, ( 3 T ξ + ξ ui vi ui vi s ( ξ ξ ξ u ' v ' u ' v ' + ( 3ξ ξ 3 s ( ξ ξ u ' v ' u ' v ' i i T i i u ξ v ξ = N 3 = ξ u v u v (7 To especially point out tat te term tat is degree of freedom of unction angle ( u', v' is te arc lengt s derivative of translational movement witout corresponding pysical angle. Te angle of linear neutral plane can be expressed as: θ = ( n u ', and te strain field can be expressed as: T T dn d N 1 + zκ dξ u z dξ ue T T s dn e s v d e N v dξ dξ e ( = [ x ', y '] + [ y ', x '] ε ξ ξ u = e ve In brief: : ε ( ξ B( be calculated as follows: and so element stiffness matrix can be gained as:, tereby te potential energy function and element stiffness witin units are able to ue Φ = s u v JB EB dzdξ (9 v 1 T T T, e e e 1 T e = s J dzdξ K B EB (1 3. Te assembling of inclusion and matrix Nowadays, in composite materials, tere are lots of models of interfacial combination of inclusions and matrixes [1]~ [18]. Tis article aims at discussing two different assembling of unites instead of interfacial model itself, and tereby use te simplest interfacial continuous displacement model to infer te constrainable relationsip between (8 175

4 degrees of freedom of inclusions and degrees of freedom of matrixes. Te infliction of tis constrainable relationsip can refer to [11]. For eac unction A of corresponding matrixes on interface, set unction B on neutral surface of inclusion in order to make te normal direction of AB along wit curve on neutral surface seem like Figure 3. Figure 3, interfacial points assignment According to continuous model of interfacial displacement, AB is always perpendicular to neutral surface and as te fixed lengt no matter before or after transformation. = ( y', x', = ( x', y' n τ, respectively is unit normal vector of inclusionary neutral surface and unit tangential vector before transformation: xm xin = n (11 After transformation: ( x + u ( x + u m m in in dy + dv ds dx + du ds =, ds ds ds ds = ( n ( u ' n τ (1 get te difference between tese two formula: um uin = ( u ' n τ (13 Te above multi-points constraint makes te displacement on interfacial unction between matrix and inclusion equal. In order to tese two fit better on te interface, we can add constrain conditions tat tangent equals between inclusion and matrix. And, usually, plane unit unctions in commercial software does not ave degrees of freedom of angle [1], matrix s tangent on interfacial points can be calculated in accordance wit isoperimetric element interpolation. Assume tat te side of unit matrix on interface corresponds to te side of isoperimetric quadrilateral η = 1: ( ξ η 1 ( + ξ η 1 ( + v // N x u, N y (14 τ m = = Te tangent of interfacial unctions between tese two units is calculated by weigted average of results gained from tese two units. Te weigt in weigted average can be considered like tat: before transformation, te tangent of matrix and inclusion on interface equals: ( β α τ τ τ (15, m1 +, m, in = After coefficient is set, we know tat, after transformation, te condition of interfacial tangent is: 1751

5 ( β α τ τ τ (16 m1 + m in = To fulfill conveniently, tis article uses multi-points constraint (13 to press conditions of border of interface. 4. Examples For given foam wit certain type included large a mount of compound materials distributing randomly in te matrix, its macroscopical attribute is determined by volume fraction of bubble and tickness of inclusion wall. By adusting tese two parameters, it is flexible to customize pysical attributes of composite materials [19][]. If we ignore te interacting influences of inclusions, we can establis a reference same as bubble s volume fraction of macroscopical composite materials and tickness of wall. And ten troug te analysis we can estimate effective modulus of macroscopical composite materials [8]. In order to assure te correctness of tis model, we first calculate te volume fraction of various inclusions and effective modulus under various tickness of wall, and ten contrast it wit traditional model. Te traditional model uses Ansys 1. and PLANE4 as unit; so tis article still uses Ansys basically as te model and ten educes stiffness. And also te inclusion assembles and connects stiffness in te way tat Section and 3 perform, and ten uses optimizer called HSL-MA57 to calculate. Example 1: as indicated in Figure 4, te size of section of compound materials is1 1, including round vacuoles wit volume ratio equaling µ. Te tickness of inclusion wall is. Te Young modulus of matrix materials and inclusion materials are respectively 1Mpa and 15Mpa, and Poisson s ratios are respectively.4 and.3. Figure 4,mesing plan of te proposed model ( µ =.3, = 1 As te contrast of tis article s case, using Ansys traditional programs needs to adopt a more dense mes to properly express inclusion of deformation. As indicated in Figure 5: Figure 5,mesing plan of te solid model ( µ =.3, = 1 By calculating te lengt of material s side wile stretcing, tension needs to be pressed and ten equivalent tensile 175

6 modulus of composite materials can also be acieved. Under tis condition, te modulus sould be: ( 1 ν σ E = (17 ε Te result tat acieved from equivalent modulus wit te cange of cavity volume fraction is sown in te following figure: Figure 6,Effective modules (=.5 Figure 7,Effective modules (=1. From Figure 6 and Figure 7, we can see tat bot of tese two calculating results are very close. It matces te expectation tat te equivalent modulus increases wile te tickness of inclusions increases and te equivalent modulus decreases wile volume fraction of vacuole increases. From te figure we can also find tat te results of te two models fit better wen mixed tickness is small, wic also matces te assumption of Kircoff-Love s tin-wall teorem. Terefore, te calculation sceme tis paper proposes is valid. We can also find tat, under comparatively larger volume ratio of cavum, te result of equivalent modulus tis paper calculates is a little bit larger tan result plane unit calculates. Tere are tree reasons: first, te gridding of Ansys entitative model is kind of tin; Second, tin-wall model ignores transformation in te direction of tickness; Tird, connection of multi-points constraints as stiff effect on interface. And we know tat te penomenon is more obvious under bigger volume ratio of cavum troug analysis, wic also matces te analyses of Figure 6 and Figure 7. In reality, matrix and inclusion appen to break easily wile under uniaxial stretc, sown as Figure 8.It is necessary to make researc of equivalent modulus wile aving partial damage on interface. We need to set up models more times wile adopting procedure of Ansys to calculate. Using te approac te paper describes, te process tat interface destroys, in reality, is a process of deleting multi-points constraints, wic seems muc easier. 1753

7 Figure 8, debonding wile under uniaxial stretc Example : Te former example uses µ = %, =.5 and two cases to calculate different equivalent modules under different size of destroyed interface. Te result is sown in Figure 9: Figure 9, effective modules versus debonding angles We can see tat tese two results are extremely close to eac oter, bot sowing tat te value of equivalent modulus decreases suddenly wen its deiscent angle reaces a certain point (about 4 degree; tis penomenon sows tat tis kind of composite foam is probable to lose stableness because of te partial drop of interface wile stretcing, wic matces te results of references [16][18]. Te model tis paper describes can be applied to te intimation of drop-off procedure of interface. CONCLUSION Tis paper tries to use combination of sell model on te basis of tin-wall teorem and practical model to solve plate problem of syntactic foams. Adopting sell model, in order to avoid te problem tat te sell wall is too tin, te paper adopts very tiny gridding wile establising te model. Te advantages of tis approac are few degrees of freedom and low cost. Te article uses te format of Kircoff-Love s sell model in plane strain, and infers strain expression and stiffness matrix of inclusion, wic makes programming easier by using wole rectangular coordinate system; And te article uses continuous displacement model of inclusion and matrix on interface, gaining te connecting constrainable relationsip between plate sell and entity points, wic was pressed by direct metod; Take advantage of te above approac to researc equivalent model of syntactic foams wile te volume ratio and tickness of sell wall are canging; Simulate condition of partial drop-off on interface. Examples indicate tat te calculating results of tis paper and te traditional way are very close to eac oter, wic proves to be effective. In addition, we can find tat tinner te tickness of wall is, so bigger te volume fraction of vacuole is, larger te modulus ration of inclusion and matrix is, and closer te two calculation results of tese two models. And tis is precisely te most common configuration parameter [1][][3][6] in practical application, so tis paper can be applied to statics calculations of composite foam and simulation of quasi-static degumming process. REFERENCES [1] Narkis M, Kenig S, Puterman M. Polymer Composites, 1984, 5(: [] Balc DK, Dunand DC. Acta Materialia, 6, 54(6:

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